Question

Prove that $\left(2+\sqrt{3}\right)$ is irrational.

Answer 1

Let $(2+\sqrt{3})$ be rational.
Then, both $(2+\sqrt{3})$ and 2 are rational.
∴ $\{(2+\sqrt{3})-2\}$ is rational [∵ Difference of two rational is rational]
⇒ $\sqrt{3}$ is rational.
This contradicts the fact that $\sqrt{3}$ is irrational.
The contradiction arises by assuming $(2+\sqrt{3})$ is rational.
Hence, $(2+\sqrt{3})$ is irrational.